determine whether the following statement is true or false. the graph of a rational function may intersect a…

Explanation:

Step1: Recall definition of horizontal asymptote

A horizontal asymptote of a rational function $y = f(x)$ is a horizontal line $y = L$ such that $\lim_{x\rightarrow\pm\infty}f(x)=L$. But this only describes the behavior of the function as $x$ approaches positive or negative infinity.

Step2: Provide an example

Consider the rational function $f(x)=\frac{x}{x^{2}+1}$. The horizontal asymptote is $y = 0$ since $\lim_{x\rightarrow\pm\infty}\frac{x}{x^{2}+1}=\lim_{x\rightarrow\pm\infty}\frac{\frac{x}{x^{2}}}{\frac{x^{2}}{x^{2}}+\frac{1}{x^{2}}}=\lim_{x\rightarrow\pm\infty}\frac{\frac{1}{x}}{1 + \frac{1}{x^{2}}}=0$. When $x = 0$, $f(0)=\frac{0}{0 + 1}=0$. So the graph of the function intersects the horizontal asymptote $y = 0$ at $x = 0$.

Answer:

True